1. The first equation is - 2x + 5y = 0
Second equation is [tex]y = \frac{2}{5} x[/tex]
5y = 2x
- 2x + 5y = 0
Hence, the two equations are equivalent.
2. [tex]a_{1} = 2, a_{2} = - 2[/tex]
[tex]b_{1} = -1, b_{2} = -1[/tex]
[tex]\frac{a_{1} }{a_{2}} =\frac{2}{-2} = -1[/tex]
[tex]\frac{b_{1} }{b_{2}} = \frac{-1}{-1} = 1[/tex]
[tex]\frac{a_{1} }{a_{2}} \neq \frac{b_{1} }{b_{2}}[/tex]
Hence, the equations are consistent.
3. [tex]a_{1} = 4, a_{2} = 6[/tex]
[tex]b_{1} = -1, b_{2} = -1[/tex]
[tex]\frac{a_{1} }{a_{2}} =\frac{4}{6} = \frac{2 }{3}[/tex]
[tex]\frac{b_{1} }{b_{2}} = \frac{-1}{-1} = 1[/tex]
[tex]\frac{a_{1} }{a_{2}} \neq \frac{b_{1} }{b_{2}}[/tex]
Hence, the equations are consistent.
4. Equations can be re-arranged as:
x + y - 4 = 0 and
x + y + 6 = 0
[tex]a_{1} = 1, a_{2} = 1[/tex]
[tex]b_{1} = 1, b_{2} = 1[/tex]
[tex]c_{1} = -4, c_{2} = 6[/tex]
[tex]\frac{a_{1} }{a_{2}} =\frac{1}{1} = 1[/tex]
[tex]\frac{b_{1} }{b_{2}} =\frac{1}{1} = 1[/tex]
[tex]\frac{c_{1} }{c_{2}} =\frac{-4}{6} = \frac{-2}{3}[/tex]
[tex]\frac{a_{1} }{a_{2}} = \frac{b_{1} }{b_{2}} \neq \frac{c_{1} }{c_{2}}[/tex]
Hence, the equations are inconsistent.
5. If we multiply the first equation by 4, we will get,
2y = -4x + 20 which is the second equation.
Hence, the equations are equivalent.
